Spatial Sampling Requirements

How Fine Should the Grid Be?

We have determined what region of k-space we can observe and what resolution we can achieve. The remaining question is: what spatial grid should we use for discretizing the scene? Too coarse a grid aliases high-frequency content; too fine wastes computation. The Nyquist-Shannon sampling theorem, applied to k-space, gives the answer.

Theorem: Spatial Sampling Theorem for Imaging

If the k-space coverage is bounded within [−κmax⁡/2,κmax⁡/2][-\kappa_{\max}/2, \kappa_{\max}/2] in each dimension (with κmax⁡=2⋅2π(f0+W/2)/c\kappa_{\max} = 2 \cdot 2\pi(f_0 + W/2)/\text{c} for the maximum combined wavenumber), then the imaging grid spacing must satisfy

Δp≤2πκmax⁡=λmin⁡2,\boxed{\Delta p \leq \frac{2\pi}{\kappa_{\max}} = \frac{\lambda_{\min}}{2}},

where λmin⁡=c/(f0+W/2)\lambda_{\min} = \text{c}/(f_0 + W/2) is the shortest wavelength in the system. Violating this condition causes spatial aliasing: replicas of the scene appear as ghost images.

This is the spatial analogue of the Nyquist theorem in time-frequency: you need at least two samples per wavelength in each direction to avoid aliasing. The maximum spatial frequency is κmax⁡/2π\kappa_{\max}/2\pi, so the sampling rate must be at least κmax⁡/π\kappa_{\max}/\pi samples per meter.

Proof
Apply the sampling theorem

The reflectivity c(p)c(\mathbf{p}) is band-limited in k-space to ∥κ∥≤κmax⁡\|\boldsymbol{\kappa}\| \leq \kappa_{\max}. By the Nyquist-Shannon theorem (in dd dimensions), it can be perfectly recovered from samples spaced at Δp=2π/κmax⁡=λmin⁡/2\Delta p = 2\pi/\kappa_{\max} = \lambda_{\min}/2 in each dimension. Coarser sampling aliases the components with spatial frequency above π/Δp\pi/\Delta p. ■\blacksquare

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Definition:

Angular Sampling Theorem

For a receiver array with aperture DD (total physical size), the angular sampling requirement is determined by the scene extent LL (diameter of the target region). The number of angular samples (independent Tx or Rx positions) needed is

Nangular≥2κLDπ⋅d(s,p0),N_{\text{angular}} \geq \frac{2\kappa L D}{\pi \cdot d(\mathbf{s}, \mathbf{p}_{0})},

where the factor accounts for the Fresnel zone structure. In the far field, this simplifies to: the angular spacing between adjacent Tx-Rx directions must be less than

Δθ≤λ2L,\Delta\theta \leq \frac{\lambda}{2L},

to avoid grating lobes. This is the angular analogue of the spatial Nyquist condition.

Violating the angular sampling condition creates grating lobes — periodic replicas of the image in angle, analogous to the grating lobes of a phased array with element spacing >λ/2> \lambda/2.

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Example: Imaging Grid Design for a 10 GHz System

Design the imaging grid for a system with f0=10f_0 = 10 GHz, W=200W = 200 MHz, and a target region of size 16×1616 \times 16 m centered at range 20 m. Determine the grid spacing, number of voxels, and whether the standard 128×128128 \times 128 grid from the codebase is appropriate.

Solution
Minimum wavelength

λmin⁡=c/(f0+W/2)=3×108/(10.1×109)=0.0297\lambda_{\min} = \text{c}/(f_0 + W/2) = 3 \times 10^8/(10.1 \times 10^9) = 0.0297 m ≈3.0\approx 3.0 cm.

Nyquist grid spacing

ΔpNyquist=λmin⁡/2=1.49\Delta p_{\text{Nyquist}} = \lambda_{\min}/2 = 1.49 cm. This is very fine and would require Q=(16/0.0149)2≈1.15×106Q = (16/0.0149)^2 \approx 1.15 \times 10^6 voxels — impractically large for most algorithms.

Practical grid spacing

The Nyquist limit assumes full 360°360° angular coverage. In practice, with limited aperture (θmax⁡=120°\theta_{\max} = 120°), the effective spatial bandwidth is smaller and the achievable resolution is coarser. A practical choice is to set Δp\Delta p equal to the achievable resolution:

Range: Δpr=c/(2W)=0.75\Delta p_r = \text{c}/(2W) = 0.75 m. Cross-range: Δpx=λ/(2sin⁡(60°))≈1.7\Delta p_x = \lambda/(2\sin(60°)) \approx 1.7 cm.

The 128×128128 \times 128 grid with Δ=16/128=0.125\Delta = 16/128 = 0.125 m is a reasonable compromise that resolves features at the range-resolution scale while being computationally tractable. It slightly over-samples in the cross-range direction and under-samples relative to the Nyquist limit in range.

Total voxels

Q=128×128=16,384Q = 128 \times 128 = 16{,}384 voxels, matching the standard configuration in the codebase. The sensing matrix A∈CMNK×16384\mathbf{A} \in \mathbb{C}^{MNK \times 16384} requires ∼\sim268 MB per pair at complex64 precision — manageable on a modern GPU.

Grating Lobes and the Antenna Connection

As Caire points out in his note, the spatial sampling in imaging is intimately connected to the grating-lobe phenomenon in antenna arrays. An array with element spacing d>λ/2d > \lambda/2 produces grating lobes — spurious beams at angles satisfying the array factor periodicity. Similarly, an imaging grid with spacing Δp>λmin⁡/2\Delta p > \lambda_{\min}/2 produces grating lobes in the reconstructed image — ghost replicas of the true scene.

The key insight from the 1D sinc/sampling analysis in Caire's paper: the total bandwidth determines the resolution (mainlobe width), while the sampling pattern determines the artifact structure (sidelobe/grating-lobe pattern). Irregular (non-periodic) sampling spreads grating-lobe energy over the image rather than concentrating it, which is why irregular arrays are generally preferred for imaging.

Spatial Aliasing in RF Imaging

Demonstrate how under-sampling the imaging grid creates aliasing artifacts. Compare the reconstructed image at different grid spacings to the Nyquist-sampled reference.

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🔧Engineering Note

Voxel Sizing Rules of Thumb

In practice, the voxel size is chosen as a compromise between computational cost and imaging quality:

  1. Each voxel should not span more than one range resolution cell: Δpr≤c/(2W)\Delta p_r \leq \text{c}/(2W).
  2. Each voxel should not span more than one angular resolution cell of any receiver array.
  3. The total number of voxels QQ is limited by memory and computation: the sensing matrix A∈CMNK×Q\mathbf{A} \in \mathbb{C}^{MNK \times Q} requires 8⋅MNK⋅Q8 \cdot MNK \cdot Q bytes (complex64).
  4. Adaptive refinement (coarse grid first, refine around active voxels) can reduce QQ while maintaining resolution where it matters.

For the standard CommIT configuration (f0=10f_0 = 10 GHz, W=200W = 200 MHz, Q=1282Q = 128^2), each voxel is 12.5×12.512.5 \times 12.5 cm — much larger than the Nyquist limit but matched to the range resolution. This is deliberate: finer gridding would not improve imaging with 200 MHz bandwidth.

Common Mistake: Over-Sampling Does Not Improve Resolution

Mistake:

Using an extremely fine imaging grid (many more voxels than measurements) expecting that the finer grid will produce a higher-resolution image.

Correction:

Resolution is fundamentally limited by the k-space coverage (bandwidth and aperture). Over-sampling the grid beyond the Nyquist rate creates a smoother-looking image (sinc interpolation) but does not reveal features finer than Δr\Delta r or Δx\Delta x. Worse, a very fine grid makes the inverse problem more severely underdetermined (MNK≪QMNK \ll Q) and amplifies noise. The grid should be matched to the achievable resolution — typically Δp∼Δr\Delta p \sim \Delta r in range and Δp∼Δx\Delta p \sim \Delta x in cross-range.

Quick Check

A system has f0=28f_0 = 28 GHz and W=400W = 400 MHz. The Nyquist grid spacing for the imaging grid is approximately:

λmin⁡/2≈5.3\lambda_{\min}/2 \approx 5.3 mm

λ0/2≈5.4\lambda_0/2 \approx 5.4 mm

c/(2W)=37.5c/(2W) = 37.5 cm

λ0≈10.7\lambda_0 \approx 10.7 mm

Correction:
λmin⁡/2≈5.3\lambda_{\min}/2 \approx 5.3 mm

λmin⁡=c/(28.2×109)=10.6\lambda_{\min} = \text{c}/(28.2 \times 10^9) = 10.6 mm, so Δp=λmin⁡/2≈5.3\Delta p = \lambda_{\min}/2 \approx 5.3 mm. However, the range resolution is c/(2×400×106)=37.5\text{c}/(2 \times 400 \times 10^6) = 37.5 cm, so in practice a much coarser grid matched to the range resolution is used.

Key Takeaway

The complete system design recipe for RF imaging is now clear: (1) Choose the array geometry to maximize angular diversity (k-space coverage). (2) Choose the bandwidth to achieve the desired range resolution: W=c/(2Δr)W = \text{c}/(2\Delta r). (3) Set the grid spacing to the achievable resolution (not the Nyquist limit, which is impractically fine). (4) Check the k-space coverage pattern for gaps. (5) Determine the SNR from the link budget. Every subsequent chapter in this book develops methods to solve the resulting inverse problem y=Ac+w\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}.

Resolution LimitsChapter Summary